Optimal. Leaf size=81 \[ \frac {1}{b x \sqrt [4]{a-b x^4}}-\frac {\sqrt [4]{1-\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt {b} \sqrt [4]{a-b x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {293, 319, 342,
281, 234} \begin {gather*} \frac {1}{b x \sqrt [4]{a-b x^4}}-\frac {x \sqrt [4]{1-\frac {a}{b x^4}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt {b} \sqrt [4]{a-b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 234
Rule 281
Rule 293
Rule 319
Rule 342
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a-b x^4\right )^{5/4}} \, dx &=\frac {1}{b x \sqrt [4]{a-b x^4}}+\frac {\int \frac {1}{x^2 \sqrt [4]{a-b x^4}} \, dx}{b}\\ &=\frac {1}{b x \sqrt [4]{a-b x^4}}+\frac {\left (\sqrt [4]{1-\frac {a}{b x^4}} x\right ) \int \frac {1}{\sqrt [4]{1-\frac {a}{b x^4}} x^3} \, dx}{b \sqrt [4]{a-b x^4}}\\ &=\frac {1}{b x \sqrt [4]{a-b x^4}}-\frac {\left (\sqrt [4]{1-\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {x}{\sqrt [4]{1-\frac {a x^4}{b}}} \, dx,x,\frac {1}{x}\right )}{b \sqrt [4]{a-b x^4}}\\ &=\frac {1}{b x \sqrt [4]{a-b x^4}}-\frac {\left (\sqrt [4]{1-\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {a x^2}{b}}} \, dx,x,\frac {1}{x^2}\right )}{2 b \sqrt [4]{a-b x^4}}\\ &=\frac {1}{b x \sqrt [4]{a-b x^4}}-\frac {\sqrt [4]{1-\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt {b} \sqrt [4]{a-b x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 7.72, size = 55, normalized size = 0.68 \begin {gather*} \frac {x^3 \sqrt [4]{1-\frac {b x^4}{a}} \, _2F_1\left (\frac {3}{4},\frac {5}{4};\frac {7}{4};\frac {b x^4}{a}\right )}{3 a \sqrt [4]{a-b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (-b \,x^{4}+a \right )^{\frac {5}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.07, size = 36, normalized size = 0.44 \begin {gather*} {\rm integral}\left (\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{4}} x^{2}}{b^{2} x^{8} - 2 \, a b x^{4} + a^{2}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.43, size = 39, normalized size = 0.48 \begin {gather*} \frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {5}{4}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\left (a-b\,x^4\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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